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Temperature-Dependent Refractive Index of Silicon and Germanium

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Temperature-Dependent Refractive Index of Silicon and Germanium Temperature-dependent refractive index of silicon and germanium Bradley J. Frey *, Douglas B. Leviton, Timothy J. Madison NASA Goddard Space Flight Center, Greenbelt, MD 20771 ABSTRACT Silicon and germanium are perhaps the two most well-understood semiconductor materials in the context of solid state device technologies and more recently micromachining and nanotechnology. Meanwhile, these two materials are also important in the field of infrared lens design. Optical instruments designed for the wavelength range where these two materials are transmissive achieve best performance when cooled to cryogenic temperatures to enhance signal from the scene over instrument background radiation. In order to enable high quality lens designs using silicon and germanium at cryogenic temperatures, we have measured the absolute refractive index of multiple prisms of these two materials using the Cryogenic, High-Accuracy Refraction Measuring System (CHARMS) at NASA’s Goddard Space Flight Center, as a function of both wavelength and temperature. For silicon, we report absolute refractive index and thermo-optic coefficient (dn/dT) at temperatures ranging from 20 to 300 K at wavelengths from 1.1 to 5.6 m, while for germanium, we cover temperatures ranging from 20 to 300 K and wavelengths from 1.9 to 5.5 m. We compare our measurements with others in the literature and provide temperature-dependent Sellmeier coefficients based on our data to allow accurate interpolation of index to other wavelengths and temperatures. Citing the wide variety of values for the refractive indices of these two materials found in the literature, we reiterate the importance of measuring the refractive index of a sample from the same batch of raw material from which final optical components are cut when absolute accuracy greater than ±5 x 10 -3 is desired. Keywords: Refractive index, silicon, germanium, cryogenic, infrared, refractometer, thermo-optic coefficient, CHARMS 1. INTRODUCTION Historically, few accurate refractive index measurements of infrared materials have been made at cryogenic temperatures. This has hampered the developments of many cryogenic infrared instruments, including the Infrared Array Camera (IRAC) for NASA’s Spitzer Space Telescope, for which, for design purposes, index values for its lens materials were extrapolated from literature values both in wavelength and in temperature. Such an approach leads to integration cycles which are far longer than anticipated, where best instrument performance is achieved by trial and error in many time-consuming and expensive iterations of instrument optical alignment. The Cryogenic High Accuracy Refraction Measuring System (CHARMS) has been recently developed at GSFC to make such measurements in the absolute sense (in vacuum) down to temperatures as low as 15 K with unsurpassed accuracy using the method of minimum deviation refractometry. 1,2,3 For low index materials with only modest dispersion such as synthetic fused silica, CHARMS can measure absolute index with an uncertainty in the sixth decimal place of index. For materials with higher indices and high dispersion such as Si or Ge, CHARMS can measure absolute index with an uncertainty of about one part in the fourth decimal place of index. Measurement and calibration methods used and recent facility improvements are discussed in previous papers. 4,5 Prism pairs were purchased for this study of silicon and germanium by the James Webb Space Telescope’s Near-Infrared Camera (JWST/NIRCam) and were fabricated from one boule of each respective material. Therefore, this study is not primarily an interspecimen variability study for specimens from different material suppliers or from different lots of material from a given supplier. Both silicon prisms were specified as “optical grade” silicon and both germanium prisms were specified as single-crystal germanium at the time of fabrication. The purity level for all four prisms is specified at >99.999% by the material vendor. Once again, no quantitative analysis of the relationship between crystal type or * [email protected] , phone 1-301-286-7787, FAX 1-301-286-0204 material purity and refractive index has been done during this study. The examination of such variability is being considered for future efforts. The apex angle of the prism for each material is designed so that the beam deviation angle for the highest index in the material’s transparent range will equal the largest accessible deviation angle of the refractometer, 60°. Common prism dimensions are refracting face length and height of 38.1 mm and 28.6 mm, respectively. Nominal apex angles for the prisms measured in this study are: Si – 20.0° and Ge – 17.0°. 2. PRESENTATION OF MEASURED INDEX DATA Detailed descriptions of our data acquisition and reduction processes are documented elsewhere 4, as are our calibration 5 procedures . In general we fit our raw measured data to a temperature dependent Sellmeier model of the form: m S )T( λ⋅ 2 n 2 (λ )T, −1 = i ∑ λ2 − λ2 i=1 i )T( where S i are the strengths of the resonance features in the material at wavelengths λi . When dealing with a wavelength interval between wavelengths of physical resonances in the material, the summation may be approximated by only a few 6 terms, m – typically three. In such an approximation, resonance strengths S i and wavelengths λi no longer have direct physical significance but are rather parameters used to generate an adequately accurate fit to empirical data. If these parameters are assumed to be functions of T, one can generate a temperature-dependent Sellmeier model for n( λ,T). Historically, this modeling approach has been employed with significant success for a variety of materials despite a rather serious sparsity of available index measurements – to cover a wide range of temperatures and wavelengths – upon which to base a model. One solution to the shortcoming of lack of measured index data has been to appeal to room temperature refractive index data at several wavelengths to anchor the model and then to extrapolate index values for other temperatures using accurate measurements of the thermo-optic coefficient dn/dT at those temperatures, which are much easier to make than accurate measurements of the index itself at exotic temperatures. 6 This is of course a potentially dangerous assumption, depending on the sample material in question and the required accuracy of refractive index knowledge. Meanwhile, with CHARMS, we have made direct measurements of index densely sampled over a wide range of wavelengths and temperatures to produce a model with residuals on the order of the uncertainties in our raw index measurements. For our models, we have found that 4th order temperature dependences in all three terms in each of S i 6 and λi work adequately well, as also found previously in the literature. The Sellmeier equation consequently becomes: 3 S )T( λ⋅ 2 n 2 (λ )T, −1 = i ∑ λ2 − λ2 i=1 i )T( where, These Sellmeier models are our best statistical representation of the measured data over the complete measured ranges of wavelength and temperature. All of the tabulated values for the refractive index of Si and Ge below have been 4 = ⋅ j Si )T( ∑Sij T j=0 4 λ = λ ⋅ j i )T( ∑ ij T j=0 calculated using this Sellmeier model based on our measured data using the appropriate coefficients in Tables 5 and 10. Depending on the sample material measured, the residuals of the measured values compared to the fits can be as low as several parts in the sixth decimal place of index. This level of fit quality for data obtained using CHARMS generally pertains to low index materials with moderately low dispersion such as fused silica, LiF, and BaF 2. For high index materials such as Si and Ge, residuals are higher and in the range of 1 x 10 -4 to 1 x 10 -5 depending on the wavelength and temperature of interest. 2.1 Silicon Absolute refractive indices of Si were measured over the 1.1 to 5.6 microns wavelength range and over the temperature range from 20 to 300 K for two test specimens which yielded equal indices to within our typical measurement uncertainty of +/- 1 x 10 -4 (see Table 1). Indices are plotted in Figure 1, tabulated in Table 2 for selected temperatures and wavelengths. Spectral dispersion is plotted in Figure 2, tabulated in Table 3. Thermo-optic coefficient is plotted in Figure 3, tabulated in Table 4. Coefficients for the three term Sellmeier model with 4 th order temperature dependence are given in Table 5. Table 1: Uncertainty in absolute refractive index measurements of silicon for selected wavelengths and temperatures wavelength 30 K 75 K 100 K 200 K 295 K 1.5 microns 8.61E-05 1.35E-04 1.20E-04 9.16E-05 8.67E-05 3.0 microns 5.00E-05 1.08E-04 9.14E-05 5.52E-05 4.55E-05 4.0 microns 4.94E-05 1.07E-04 9.07E-05 5.44E-05 4.47E-05 5.0 microns 5.08E-05 1.07E-04 9.00E-05 5.42E-05 4.46E-05 Absolute refractive index of silicon with temperature 3.58 50 K 3.56 3.54 100 K 3.52 150 K 3.50 200 K 3.48 295 K 3.46 3.44 3.42 absolute refractive index 3.40 3.38 0.0 1.0 2.0 3.0 4.0 5.0 6.0 wavelength [microns] Figure 1: Measured absolute refractive index of silicon as a function of wavelength for selected temperatures Table 2: Measured absolute refractive index of silicon for selected wavelengths and temperatures wavelength 30 K 40 K 50 K 60 K 70 K 80 K 90 K 100 K 150 K 200 K 250 K 295 K 1.1 microns 3.51113 3.51123 3.51146 3.51181 3.51229 3.51288 3.51358 3.51439 3.51979 3.52703 3.53555 3.54394 1.2 microns 3.49071 3.49080 3.49103 3.49137 3.49183 3.49241 3.49309 3.49387 3.49910 3.50610 3.51437 3.52253
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